Narrow Results

Average options are path-dependent and have payoffs which depend on the average price over a fixed period leading up to the maturity date. This option is of interest and important for thinly-traded assets since price manipulation is prohibited, and both the investor and issuer may enjoy a certain degree of protection from the caprice of the market. However, to deal with unexpected situations incurred the usual simple average options may not be sufficient. Therefore, in this paper, we consider to propose a more general weight instead of the simple average, for which it may be possible to control the weight in the light of the unexpected circumstances. Further, we derive approximate solutions for the weighted sums of asset prices, and in order for these formulae to be applicable some adjustment must be taken into account along with Monte Carlo simulations. Finally, some comparisons for these results are made.

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.